Thin ﬁlm dynamics on a vertically rotating disk partially ... · Thin ﬁlm dynamics on a...

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Applied Mathematical Modelling xxx (2007) xxx–xxx

www.elsevier.com/locate/apm

Thin film dynamics on a vertically rotating diskpartially immersed in a liquid bath

K. Afanasiev a, A. Munch b, B. Wagner a,*

a Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germanyb Institute of Mathematics, Humboldt University, 10099 Berlin, Germany

Received 1 April 2006; received in revised form 1 June 2007; accepted 5 June 2007

Abstract

The axisymmetric flow of a thin liquid film is considered for the problem of a vertically rotating disk that is partiallyimmersed in a liquid bath. A model for the fully three-dimensional free boundary problem of the rotating disk, that drags athin film out of the bath is set up. From this, a dimension-reduced extended lubrication approximation that includes themeniscus region is derived. This problem constitutes a generalization of the classic drag-out and drag-in problem to thecase of axisymmetric flow. The resulting nonlinear fourth-order partial differential equation for the film profile is solvednumerically using a finite element scheme. For a range of parameters steady states are found and compared to asymptoticsolutions. Patterns of the film profile, as a function of immersion depth and angular velocity are discussed.� 2007 Elsevier Inc. All rights reserved.

1. Introduction

The problem of rotating thin film flows has been investigated extensively, both theoretically and experimen-tally, due to the many technological applications. Starting with the work by Emslie et al. [1], various aspects ofthis type of surface tension driven flow, influencing the shape and stability of the film, have been studied.These include for example non-Newtonian effects the paper by [2], for evaporation [3] and Coriolis force[4,5], see also [6] for a recent experimental study. As with those studies most of them dealt with a configurationwhere the fluid layer is moving on a horizontally rotating disk. Making use of the large scale separationbetween the small thickness of the film and the length scale of the evolving patterns, thin film models whereused to derive dimension-reduced models of the underlying three-dimensional free boundary problems.

For the situation of a disk that is partially immersed in a bath of liquid and rotating about the horizontalaxis, thereby dragging out a thin film onto the disk (see Fig. 1), there are far fewer studies even though thisconfiguration is typical for many applications. For example, for oil disk skimmers, which is used as an effectivedevice for oil recovery and as an alternative to toxic chemical dispersants, used after an offshore oil spill.

0307-904X/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2007.06.020

* Corresponding author.E-mail address: [email protected] (B. Wagner).

Please cite this article in press as: K. Afanasiev et al., Thin film dynamics on a vertically rotating disk ..., Appl. Math.Modell. (2007), doi:10.1016/j.apm.2007.06.020

mailto:[email protected]

a

Liquid

r

z

R

θ

Γ

R in

out

pool

Ω

Fig. 1. Configuration of a single disk within a PET-reactor.

2 K. Afanasiev et al. / Applied Mathematical Modelling xxx (2007) xxx–xxx

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Another application is the fluid dynamical aspects in connection with the synthesis of polyethylenterephthalat(PET) in polycondensation reactors. These typically consist of a horizontal cylinder that is partially filled withpolymer melt and contains disks rotating about the horizontal axis of the cylinder, thus picking up and spread-ing the melt in form of a thin film over a large area of the disks.

These type of problems always involve a meniscus region, that connects the thin film to the liquid bath anda spacially oscillating region shortly before the film is dragged into the bath again. In these two regions thescale separation is not large anymore and hence the lubrication approximation is not valid there. Nevertheless,the meniscus does play the crucial role of fixing the height of the dragged out film and therefore both, themeniscus region and similarly the drag-in region that connecting the thin film to the liquid bath must beaccounted for in a dimension-reduced model, which we will derive in this paper. The classic and far simplersetting of the free boundary problems for falling and rising thin film flows on vertical as well as inclined planeshas been investigated as early as in the work by Landau and Levich [7]. Their work lead to the prediction ofthe height and shape of the thin film emerging out of the meniscus. The results were improved by Wilson [8]and for the case of a Marangoni-driven rising film by [9] and Munch [10], using systematic asymptotic analysisin the limit of small capillary numbers. Such asymptotic analyses can be applied to more complex situationssuch as the problem we consider here for the vertically rotating disk.

Previous studies for this problem was performed by Christodoulo et al. [11]. The employed the analysis ofthe meniscus region by Wilson [8] for the problem of flow control for rotating oil disk skimmers. Their studydid not extend further into the thin film region on the remainder of the disk. However, for many applications itis important to answer questions for example on the maximum surface area of the film profile or the optimumvolume of liquid that is dragged out and spread on the disk, for which the predictions resulting from the sim-ple drag-out study will not be sufficient. Up to now no complete model for the vertically rotating disk, includ-ing its numerical solution has appeared. This will be the topic of this paper.

In Section 2 we set up the corresponding three-dimensional free boundary problem. A fully three-dimen-sional analysis of such flows represents a very time consuming task, analytically and numerically. To be ableto perform systematic parameter studies we therefore exploit the large separation of scales to obtain a dimen-sion-reduced lubrication model. This model will then be extended to match to the flow field in the meniscusregion. For the resulting model we develop in Section 3 a weak formulation and a corresponding finite elementdiscretization for the full dynamical problem. Since we want to address here issues of maximum surface area ofthe film profile or the optimum volume of liquid that is dragged out, we are interested mainly in the long-timebehavior. This will be the focus of the following sections. In Section 4 we solve and discuss for a range ofparameters the emerging steady state solutions. Their shapes near the meniscus region are then compared


K. Afanasiev et al. / Applied Mathematical Modelling xxx (2007) xxx–xxx 3

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to the asymptotic solution of the corresponding drag-out problem. Away from the liquid bath the film profileis compared to asymptotic solutions, by using the methods of characteristics. For both regions excellent agree-ment is found. Finally, we discuss the novel patterns for the film profile as the immersion depth, or the angularvelocity is varied.

2. Formulation

2.1. Governing equations

We consider the isothermal flow of an incompressible, viscous liquid on a vertical disk rotating in the ver-tical plane and partially immersed in the liquid. We assume that the disk of radius R rotates with the angularvelocity X about a horizontal axis, which has distance a to the bath, see Fig. 1.

To formulate the problem, we introduce cylindrical polar coordinates ðr; h; zÞ in the laboratory frame ofreference. We let the liquid velocity vector have components ðu; v;wÞ and let x denote the angular velocityvector with components ð0; 0;XÞ. The momentum balance equations can be expressed as

PleaMod

q ut þ uur þvr

uh �v2

rþ wuz

� �¼ �pr þ l Du� 2vh

r2� u

r2

� �� qg sin h; ð2:1aÞ

q vt þ uvr þvr

vh þuvrþ wvz

h i¼ � ph

rþ l Dvþ 2uh

r2� v

r2

� �� qg cos h; ð2:1bÞ

q wt þ uwr þvr

wh þ wwz

h i¼ �pz þ lDw; ð2:1cÞ

where

Df ¼ 1

rðrfrÞr þ

fhh

r2þ fzz: ð2:2Þ

We let q, l and p denote the density, dynamic shear viscosity and the pressure of the liquid, respectively. Theexternal force here is gravity and g denotes the gravitational constant.

The continuity equation is

1

rðruÞr þ

1

rvh þ wz ¼ 0: ð2:3Þ

For the boundary condition at the surface of the disk, i.e. z = 0, that rotates with the velocity X, we impose theno-slip condition for u and v and the impermeability condition for w. Hence, we have

u ¼ 0; v ¼ rX; w ¼ 0; ð2:4Þ

respectively.At the free boundary z ¼ hðr; h; tÞ we require the normal stress condition

nPn ¼ 2rj; ð2:5Þ

the tangential stress conditions

nPti ¼ 0; where i ¼ 1; 2 ð2:6Þ

and the kinematic condition

ht ¼ w� ujhhr �1

rvjhhh; ð2:7Þ

which can also be written, upon using the continuity equation, as

ht ¼ �1

ro

orrZ h

0

udz� 1

ro

oh

Z h

0

vdz: ð2:8Þ

se cite this article in press as: K. Afanasiev et al., Thin film dynamics on a vertically rotating disk ..., Appl. Math.ell. (2007), doi:10.1016/j.apm.2007.06.020


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The normal and the tangential vectors in radial and angular direction are given by

PleaMod

n ¼ ð�hr;�hh=r; 1Þð1þ h2

r þ h2h=r2Þ1=2

; t1 ¼ð1; 0; hrÞ

ð1þ h2h=r2Þ1=2

; t2 ¼ð0; 1; hh=rÞð1þ h2

h=r2Þ1=2; ð2:9Þ

respectively. The stress tensor P is symmetric and has the components

Prr ¼ �p þ 2lur; Phh ¼ �p þ 2lvh

rþ u

r

� �; Pzz ¼ �p þ 2lwz;

Prh ¼ luh

rþ vr �

vr

� �; Phz ¼ l vz þ

wh

r

� �; Prz ¼ lðwr þ uzÞ:

ð2:10Þ

Finally, we assume surface tension to be constant and denote it by r and the mean curvature is given by

j ¼ 1

2

1

ro

orrhr

ð1þ h2r þ h2

h=r2Þ1=2þ 1

ro

ohhh=r

ð1þ h2r þ h2

h=r2Þ1=2

!: ð2:11Þ

Using this in Eqs. (2.5) and (2.6) we obtain the boundary conditions for the normal stress

�p þ 2l

1þ h2r þ h2

h=r2

uh

rþ vr �

vr

� � hrhh

r� ðwr þ uzÞhr � vz þ

wh

r

� � hh

rþ urh

2r þ ðvh þ uÞ h

2h

r3þ wz

� �

¼ r1

ro

orrhr

ð1þ h2r þ h2

h=r2Þ1=2þ 1

ro

ohhh=r

ð1þ h2r þ h2

h=r2Þ1=2

" #; ð2:12Þ

the tangential stress condition in radial direction

2ðwz � urÞhr �uh

rþ vr �

vr

� � hh

rþ ðwr þ uzÞð1� h2

r Þ � vz þwh

r

� � hrhh

r¼ 0 ð2:13Þ

and the tangential stress condition in angular direction

2 wz �vh

r� u

r

� � hh

r� uh

rþ vr �

vr

� �hr þ vz þ

wh

r

� �1� h2

h

r2

� �� ðwr þ uzÞ

hrhh

r¼ 0: ð2:14Þ

2.2. Lubrication approximation

The solution of the above three-dimensional free boundary problem represents, analytically and numeri-cally a time consuming task for making accurate parameter studies. The key idea that we make use of herein order to obtain a mathematically and numerically tractable problem, is the exploitation of the scale sepa-ration in most parts of this flow problem.

We begin by introducing dimensionless variables and set

r ¼ L�r; h ¼ �h; z ¼ H�z; u ¼ U�u; v ¼ U�v; w ¼ W �w; p ¼ P�p; t ¼ T �t: ð2:15Þ

The characteristic velocity U is set by the velocity of the rotating disk. For given radius R of the disk we let

U ¼ RX: ð2:16Þ

We determine the scale for the characteristic height H by balancing the dominant viscous term with gravita-tional term in the u-momentum equation, which yields

H ¼ffiffiffiffiffiffiffilUqg

s: ð2:17Þ

Furthermore, we require that the pressure must also balance the dominant viscous term, so that

P ¼ lUL

H 2ð2:18Þ



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and that surface tension is important, so that from the normal stress boundary condition we find

PleaMod

P ¼ rH

L2: ð2:19Þ

This yields the scale for L as

L ¼ HlUr

1=3ð2:20Þ

and the time scale is fixed by T ¼ L=U .We assume that the liquid film is very thin and that the velocity in the direction normal to the disk is much

smaller than along the disk. We let

e ¼ HL� 1 ð2:21Þ

be a small parameter and W ¼ eU . Note that this also means that the capillary number Ca is small,

Ca1=3 ¼ lUr

� �1=3

¼ HL� 1: ð2:22Þ

With these scales the non-dimensional equations are

e2Re ut þ uur þvr

uh �v2

rþ wuz

� �¼ �pr þ uzz � sin hþ e2 ðrurÞr

rþ uhh

r2� 2vh

r2� u

r2

� �; ð2:23aÞ

e2Re vt þ uvr þvr

vh þuvrþ wvz

h i¼ � ph

rþ vzz � cos hþ e2 ðrvrÞr

rþ vhh

r2þ 2uh

r2� v

r2

� �; ð2:23bÞ

e4Re wt þ uwr þvr

wh þ wwz

h i¼ �pz þ e2wzz þ e4 ðrwrÞr

rþ whh

r2

� �; ð2:23cÞ

where the Reynolds number is Re ¼ qUL=l and we have dropped the ‘�’s.The boundary conditions at the disk, z = 0 are

u ¼ 0; v ¼ ar; w ¼ 0; ð2:24Þ
where a ¼ L=R.
The boundary conditions at the free liquid surface z ¼ hðr; h; tÞ are the conditions for normal and tangentialstresses

� pþ 2e2

1þ e2h2r þ e2h2

h=r2e2 uh

rþ vr �

vr

� �hrhh

r

��ðe2wr þ uzÞhr � vz þ e2 wh

r

� �hh

rþ e2urh

2r þ e2ðvh þ uÞh

2h

r3þwz

�¼ 1

ro

orrhr

ð1þ e2h2r þ e2h2

h=r2Þ1=2þ 1

ro

ohhh=r

ð1þ e2h2r þ e2h2

h=r2Þ1=2

" #; ð2:25Þ

2e2ðwz � urÞhr � e2 uh

rþ vr �

vr

� �hh

rþ ðe2wr þ uzÞð1� e2h2

r Þ � e2 vz þ e2 wh

r

� �hrhh

r¼ 0; ð2:26Þ

2e2 wz �vh

r� u

r

� �hh

r� e2 uh

rþ vr �

vr

� �hr þ vz þ e2 wh

r

� �1� e2 h2

h

r2

� �� e2ðe2wr þ uzÞ

hrhh

r¼ 0; ð2:27Þ

and the kinematic boundary condition

ohot¼ � 1

ro

orrZ h

0

udz� �

� 1

ro

oh

Z h

0

vdz� �

: ð2:28Þ

2.3. Region near the liquid bath

The scalings introduced so far are appropriate for the thin film region away from the liquid bath. Thisyields a leading order theory that retains the terms that are dominant for the film profile on the disk, whereslopes are small. Towards the liquid bath the film profile becomes (in fact infinitely) steep and a lubrication



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scaling is no longer appropriate. Rather, the profile is governed by the balance of gravity and surface tensionforces, in fact, much as in a static meniscus. Hence, the appropriate length scales for all spatial coordinates isthe capillary length scale lcap ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffir=ðqgÞ

p.

This length scale can be easily expressed in terms of the lubrication length scales H and L times an appro-priate power of e, so that the new meniscus length scales (denoted by tildes) become

PleaMod

eH ¼ e�3=2H ; eL ¼ e�1=2L: ð2:29Þ
The parallel velocity scale is unchanged and equal to U ¼ RX, while the normal is now U instead of eU . Thetime scale
eT ¼ eLU¼ e�1=2T ð2:30Þ

is again a result of the kinematic condition. The pressure scale is determined by surface tension, and we find
eP ¼ ffiffiffiffiffiffiffiffirqgp ¼ e�1=2P : ð2:31Þ
Hence, all variables can be transformed to meniscus scalings simply by rescaling with powers of e, according to

r ¼ e�1=2~r; z ¼ e�3=2~z; h ¼ e�3=2~h; u ¼ ~u; v ¼ ~v; w ¼ e�1 ~w; t ¼ e�1=2~t; p ¼ e�1=2~p: ð2:32Þ
Inserting these scalings into (2.23a) and (2.28), yields the rescaled equations:
e3Re ut þ uur þvr

uh �v2

rþ wuz

� �¼ �pr þ e3uzz � sin hþ e3 ðrurÞr

rþ uhh

r2� 2vh

r2� u

r2

� �; ð2:33aÞ

e3Re vt þ uvr þvr

vh þuvrþ wvz

h i¼ � ph

rþ e3vzz � cos hþ e3 ðrvrÞr

rþ vhh

r2þ 2uh

r2� v

r2

� �; ð2:33bÞ

e4Re wt þ uwr þvr

wh þ wwz

h i¼ �pz þ e3wzz þ e3 ðrwrÞr

rþ whh

r2

� �; ð2:33cÞ

where the Reynolds number is Re ¼ qUL=l ¼ e1=2qUeL=l ¼ e1=2 ~Re and where we have dropped the ‘~’s.The boundary conditions at the disk, z = 0 are

u ¼ 0; v ¼ ar; w ¼ 0; ð2:34Þ
where a ¼ ‘cap=R.
The boundary conditions for normal and tangential stresses become at z ¼ hðr; h; tÞ:

� p þ 2e3

1þ h2r þ h2

h=r2

uh

rþ vr �

vr

� � hrhh

r� ðwr þ uzÞhr � vz þ

wh

r

� � hh

rþ urh

2r þ ðvh þ uÞ h

2h

r3þ wz

� �¼ 1

ro

orrhr

1þ h2r þ h2

h=r2 1=2

þ 1

ro

ohhh=r

ð1þ h2r þ h2

h=r2Þ1=2

" #; ð2:35Þ

2ðwz � urÞhr �uh

rþ vr �

vr

� � hh

rþ ðwr þ uzÞð1� h2

r Þ � vz þwh

r

� � hrhh

r¼ 0; ð2:36Þ

2 wz �vh

r� u

r

� � hh

r� uh

rþ vr �

vr

� �hr þ vz þ

wh

r

� �1� h2

h

r2

� �� ðwr þ uzÞ

hrhh

r¼ 0: ð2:37Þ

We now retain all terms that appear to leading order either in the lubrication or the meniscus scalings. Notethat, in the meniscus scalings, the velocity field decouples to leading order from the pressure field that deter-mines the surface profile. Hence the dominant terms that govern h in these scalings consists of the pressure andgravity terms, and of surface tension, based on the full nonlinear expression for curvature. All these terms al-ready appear also in the lubrication scaling, except for the nonlinear curvature. Hence our approximate modelretains essentially the terms from a leading order lubrication theory and the nonlinear curvature term, i.e., inthe bulk we have,

0 ¼ �pr þ e3uzz � sin h; 0 ¼ � ph

rþ e3vzz � cos h; 0 ¼ �pz: ð2:38Þ



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Boundary conditions at z = 0 are given by (2.34), and at z ¼ h:

PleaMod

�p ¼ 1

ro

orrhr

ð1þ h2r þ h2

h=r2Þ1=2þ 1

ro

ohhh=r

ð1þ h2r þ h2

h=r2Þ1=2

" #; uz ¼ 0; vz ¼ 0: ð2:39Þ

Integrating first pz ¼ 0 yields a solution that does not depend on z, and the parallel components for the veloc-ity can easily be found to be

u ¼ e�3ðpr þ sin hÞðz2=2� hzÞ; v ¼ e�3ðph=r þ cos hÞðz2=2� hzÞ þ ar: ð2:40Þ
We plug this into the mass conservation relation (2.8)
ht ¼ �1

ro

orrZ h

0

udz� 1

ro

oh

Z h

0

vdz

and obtain, after rescaling time according to t ¼ ��3t0 (dropping the prime):

ht ¼1

ro

orr

h3

3ðpr þ sin hÞ

� �þ 1

ro

ohh3

3ðph=r þ cos hÞ � bXrh

� �; ð2:41Þ

where we have introduced X ¼ lX=ffiffiffiffiffiffiffiffiqgrp

.Far away from the disk we expect the liquid to be at rest, since the liquid flow diminishes. The shape of the

liquid is governed by the hydrostatic balance, so that the liquid surface is flat, i.e. its curvature is zero, and it isorthogonal to the direction of the gravitational force. Therefore we require that the function h that describesthe surface tends to infinity as the far field level of the reservoir is approached (r! �a= sin h), and the non-linear curvature tends to zero. Therefore, we have

hðr; hÞ ! 1; pðr; h; tÞ ! 0 as r! �a= sin h: ð2:42Þ
Note at this point, that for the numerical simulation, the domain has to be cut-off at a slightly higher levelr ¼ �a�= sin h. In the numerics, we enforce zero curvature at the cut-off of the domain, i.e., set p = 0 there.
Furthermore, the computational domain is intended to be an approximation of a very large reservoir,which quickly equilibrates any mass change that occurs if liquid is transferred onto or from the disk. We cap-ture this by setting h to a fixed value (h = 1) at the cut-off r ¼ �a�= sin h. We checked that the results of thesimulations were converged, i.e. changing the value of h at the cut-off did not affect the results significantly.Summarizing, we use the following conditions for the cut-off domain:

h ¼ 1; p ¼ 0 at r ¼ �a�= sin h: ð2:43Þ
We remark here, that in a converged result, a� and a are very close values, so that in the following presentationof the results, we will not distinguish these two quantities, and drop the �.
For the boundary conditions towards the inner (r ¼ Rin) and outer (r ¼ Rout) confinements of the disk weassume that both, the reservoir and the disk to be enclosed in a cylinder that tightly surrounds the disk, whichis an arrangement that reflects the typical situation inside a PET reactor. No liquid is injected or lost at the(solid) axis, nor through the cylinder; therefore, we set the mass flux to zero, which is the significance ofthe following boundary conditions

pr þ sin h ¼ 0; as r! Rin;Rout: ð2:44Þ
A second condition is required. At the axis, the liquid surface meets the solid axis, forming a contact-line,where it is natural to impose a contact angle condition. We set the contact angle to 90�, i.e. hr ¼ 0, whichis the easiest value to implement in our finite element approach. Hence, we have
hr ¼ 0; as r! Rin;Rout: ð2:45Þ

3. Numerical method

We now give a brief description of the numerical method used to solve problems (2.41) and (2.45). Themeniscus equations may be rewritten for simplicity as



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PleaMod

rohot¼ oQr

orþ oQh

oh; ð3:1Þ

� 1

2rp ¼ oqr

orþ oqh

oh; ð3:2Þ

where fluxes Qr; qr and Qh; qh in r and h directions are defined as

Qr ¼ rh3

3ðpr þ sin hÞ; qr ¼ rhrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ h2r þ h2

h=r2

q ; ð3:3Þ

Qh ¼ h3

3

1

rph þ cos h

� �þ rXh; qh ¼ hh

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h2

r þ h2h=r2

q ; ð3:4Þ

respectively. For the outlet boundary condition we take natural boundary condition, i.e. zero fluxes in thedirection of a normal vector.

Qrðr; h; tÞ ¼ 0; r! Rout; ð3:5Þqrðr; h; tÞ ¼ 0; r! Rout: ð3:6Þ

Similarly, we choose for the conditions towards the origin the natural boundary conditions

Qrðr; h; tÞ ¼ 0; r! Rin; ð3:7Þqrðr; h; tÞ ¼ 0; r! Rin: ð3:8Þ

For the immersing boundary condition, where the thin film connects to the liquid bath we let the curvature ofthe free surface vanish. Hence, we require the boundary conditions (2.42) and (2.43).

The weak formulation for (3.1) and (3.2) under the boundary conditions (3.5), (3.8) and (2.42), (2.43) can bederived by multiplying (3.1) and (3.2) by a suitable test function /, integrating over the domain K ¼ fðr; hÞ :Rin < r < Rout and r sin h > �ag and evaluating at the boundary. Then the weak formulation of the boundaryvalue problem for (3.1,3.2) requires us to seek ðh; pÞ 2 H 1ðKÞ, such that
Z
Krohot

/dK ¼ �Z

KQr o/

orþ Qh o/

oh

� �dKþ

ZC

Qr/nr dC; ð3:9Þ

1

2

ZK

rp/dK ¼Z

Kqr o/

orþ qh o/

oh

� �dK�

ZC

qr/nr dC ð3:10Þ

for all functions / 2 V ¼ W 12ðKÞ. Respecting the boundary conditions pr ¼ 0; hr ¼ 0, the following integral

equations
ZK
rohot

/dK ¼ �Z

KQr o/

orþ Qh o/

oh

� �dKþ

ZC

rh3

3B cos h/

� �dC; ð3:11Þ

1

2

ZK

rp/dK ¼Z

Kqr o/

orþ qh o/

oh

� �dK ð3:12Þ

will now be discretised. For the discretisation of the problem we divide the domain K in non-overlapping tri-angular elements Ke and replace H 1ðKÞ and V ðKÞ by finite dimensional subspaces S and V h, respectively. Wealso choose / ¼ /i; i ¼ 1; 2; . . . ;N with N denoting the number of nodes in the element Ke and let

heðr; h; tÞ ¼Xi¼1;N

hiðtÞ/iðr; hÞ; ð3:13Þ

peðr; h; tÞ ¼Xi¼1;N

piðtÞ/iðr; hÞ; ð3:14Þ

be the functions that approximate h and p on this element, respectively. The domain integrals can now be re-placed by the sum of integrals taken separately over the elements of triangulation.

The details of the finite element scheme is described in the following section.



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3.1. Finite element scheme

Let the time interval ½0; T � be subdivided into intervals with the time step s, tn ¼ tn�1 þ s; n ¼ 1; 2; . . . ;NT

and denote

PleaMod

hn ¼

h1ðtnÞh2ðtnÞ

..

.

hN ðtnÞ

0BBBB@1CCCCA; pn ¼

p1ðtnÞp2ðtnÞ

..

.

pNðtnÞ

0BBBB@1CCCCA:

By substitution of Eqs. (3.13) and (3.14) into the weak formulation and its implicit backward Euler discreti-sation, expressions (3.11) and (3.12) can be written in matrix notation as the following finite nonlinear system

Lhnþ1 þ s½Crgh1ðhnþ1; pnþ1Þ þ Chgh

2ðhnþ1; pnþ1Þ þ sðhnþ1Þ� ¼ Lhn; ð3:15ÞLpnþ1 ¼ 2½Crgp

1ðhnþ1Þ þ Chgp2ðhnþ1Þ�; ð3:16Þ

where matrices and vectors are defined by

Lij ¼Z

Ke

r/i/j dK; ð3:17Þ

Crij ¼

ZKe

o/i

or/j dK; ð3:18Þ

Chij ¼

ZKe

o/i

oh/j dK; ð3:19Þ

Mij ¼Z

Ke

/i/j dK; ð3:20Þ

gh1 ¼

rw3

qpr þ B cos h

; ð3:21Þ

gh2 ¼

w3

qph

r� B sin hþ rKh

� �; ð3:22Þ

gp1 ¼

rqhr

ð1þ ðqhr Þ

2 þ ðqhhÞ

2=r2Þ

12

; ð3:23Þ

gp2 ¼

qhh

r2ð1þ ðqhr Þ

2 þ ðqhhÞ

2=r2Þ

12

; ð3:24Þ

w ¼M�1a; ð3:25Þ

ai ¼Xm;l;j

hmhlhj

ZK

/m/l/j/i dK; ð3:26Þ

Y r ¼M�1ðCrÞT ; Y h ¼ M�1ðChÞT ; ð3:27Þqh

r ¼Y rh; qhh ¼ Y hh; ð3:28Þ

qpr ¼Y rp; qp

h ¼ Y hp; ð3:29Þ

si ¼0; Ce :¼ Ke

TC ¼ 0;R

Ce

h3i

3Br cos h

� �dC; Ce 6¼ 0:

(ð3:30Þ

Evaluation of matrix and vector coefficients: The various element matricies and vectors expressed by theequations above are spatial integrals of the various interpolation functions and their derivatives. These inte-grals can be evaluated analytically. The remaining ones are obtained using numerical quadrature procedure.Matrix and vector coefficients for triangular elements are evaluated using a seven-point quadrature scheme forquadratic triangles.


φφφφφφ

1 = L 1(2L 1 1)

2 = L 2(2L 2 1)

2 = L 3(2L 3 1)

4 = 4L 1L 2

5 = 4L 2L 3

6 = 4L 3L 141

6

3

5

_

_

_

2

Fig. 2. Basic functions for a six node quadratic triangular element.


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Triangulation: We use the six node quadratic triangular elements as shown in Fig. 2 and following basicfunctions written in the so called natural coordinates Li; i ¼ 1; 2; 3 based on area ratios (see in [12]). The gridswere generated by using the automatic mesh generator [13] based upon the Delaunay refinement algorithm.

Assembling the global equation system: The contributions of the element coefficient matrices and vectors(3.17), (3.30) are added by the common global node for the assembling of the global nonlinear equation systemsimilar to [14]. The global equation system can be written in the form

PleaMod

RðUÞ ¼ F ; ð3:31Þ

where U is constructed from the vectors hnþ1 and pnþ1 in all grid nodes.Time stepping: The time stepping algorithm is customarily implemented with a Newton–Raphson equilib-

rium iteration loop. In the each time step the following nonlinear problem must be solved

GðUÞ :¼ RðUÞ � F ¼ 0: ð3:32Þ

The linearized equation can be written on the basis of the Taylor expansion

GðUiþ1Þ ¼ GðU iÞ þoGoU

��U¼Ui|fflfflfflffl{zfflfflfflffl}

KðUiÞ

DU iþ1:

At each step of Newton’s method, some direct or iterative method must be used to solve the large linear alge-bra problem produced by the two-dimensional linearized operator

KðU iÞDUiþ1 ¼ �GðU iÞ; ð3:33Þwith U iþ1 ¼ U i þ DU iþ1: ð3:34Þ

Here, we find it convenient to use the non-symmetric multi-frontal method for large sparse linear systems fromthe packet UMFPACK [15].

4. Steady states

4.1. The half-immersed disk

We first discuss the case for the half-immersed disk at some length and compare our numerical results toasymptotic solutions near the liquid bath and in the thin film region of the disk before we consider differentimmersion depths.

4.1.1. Numerical results

We consider now a disk rotating about the horizontal axis with a constant angular speed X and being half-immersed in the liquid bath. The triangulation of the computational domain is performed in cylindrical coor-dinates r; h. The finite element mesh, used here, consists of 9533 triangular elements and 19522 nodes. Themesh is refined on the boundary Cpool to resolve the meniscus region. The steady state for Eqs. (3.1, 3.2) is



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obtained via time integration with an adaptive time step. As the stopping criterion for the Newton iterations ageneral threshold for the residuum jjGðUÞjj < 10�13 is applied.

Without loss of generality we choose the following values for the parameters throughout the paper:l ¼ 1 Pa s, q = 1000 kg/m3, r ¼ 72:7e� 3 N=m, R ¼ 2:723e� 2 m, g = 9.81 m/s2. For the initial state wechoose a partially constant profile on the top of the disk and a partially parabolic one towards the liquid bath,as shown in the Fig. 3. For the given U ¼ 7:917 e� 4 m=s and R we find the angular velocityX ¼ U=R ¼ 0:02908 s�1 ¼ 0:277 rpm, where the last equality is obtained by multiplying with 60=2p. For thesevalues the capillary number is small Ca = 0.01089, the length scales are eH ¼ eL ¼lcap ¼ ðr=qgÞ1=2 ¼ 2:723e� 3 m, bX ¼ 1:089e� 3 and the dimensionless radius of the disk is 10. The resultingsteady state is shown on the top of Fig. 3. It shows the an increase of thickness as the radius and hence theangular velocity increases, as expected. With increasing h gravity causes the film to move downwards resultingin a ridge of fluid, that thickens with increasing h and reenters the bath with a typical capillary ridge. If weincrease X, while keeping the other dimensional parameters and disk radius fixed, neither the length scalesnor the dimensionless radius of the disk change, but U, the capillary number and bX do. As examples we let

PleaMod

X ¼ 1:0 rpm:U ¼ 0:2851e� 2 m=s; Ca ¼ 0:03922; bX ¼ 3:922e� 3;

X ¼ 2:0 rpm:U ¼ 0:5703e� 2 m=s; Ca ¼ 0:07845; bX ¼ 7:843e� 3:

Fig. 4 illustrates the steady states for rotation velocities X = 0.277, 1.0, 2.0 rpm from top to bottom, respec-tively. In all cases the height of the film in the figures is multiplied by the factor of 10 to contrast more clearlythe structure of the film patterns.

One observes for all values of X of the steady solutions a region of liquid drag-out with a meniscus profileand a drag-in region with a capillary wave on the opposite side of the axis. Such an oscillation of the height istypically found for the reverse Landau–Levich problem when a liquid thin film is dragged into a liquid bath,see for example [16–19]. It can be seen more clearly when comparing the cross sections of the liquid profiles atconstant radii. In Fig. 5 we compare for the radius r = 9 the cross section for X = 0.277, 1.0, 2.0, 3.0 rpm.(Note, that here as further below, values such as for r without an explicit dimensions are in fact dimensionless.)The figure also shows that the average liquid height increases when X increases.

These results are qualitatively in accordance with the problem for the drag-out and drag-in cases. We willfurther investigate the quantitative comparison.

4.1.2. Comparison with the drag-out problem

Asymptotic estimate of the film thickness: We now derive an asymptotic approximation of the film thicknessusing a one-dimensional approximation based on the results of Landau, Levich [7] and Wilson [8] for the pla-nar-symmetric case.

0 6 8y

0

0.005

0.01

0.015

0.02

h

2 4

Fig. 3. Initial state.


Fig. 4. Steady solutions at X = 0.277, 1.0, 2.0 rpm.


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For our comparisons we focus on the case where the disk is half-immersed, i.e. a = 0. Then, if we onlyretain the axial components in the stationary form of (2.41), and after substituting rh 7! y, r dh 7! dy we obtainthe equation

PleaMod

d

dyh3

3ðpy þ 1Þ � bXrh

� �¼ 0; ð4:1Þ


5 10 15 20 25 30Arc length

0

0.05

0.1

0.15

0.2

h

r = 9, Ω = 0. 277 rpmr = 9, Ω = 1.0 rpmr = 9, Ω = 2.0 rpmr = 9, Ω = 3.0 rpm

28 29 30Arc length

0.02

0.04

0.06

0.08

h


Fig. 5. Comparisons of the profiles of the cross sections for the film profile for radius r = 9. On the right hand side where the film is pulledinto the bath a capillary wave is formed. This region is enlarged on the right figure. Note, ‘‘Arc length’’ denotes rh.


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with

PleaMod

p ¼ � d

dyhy

ð1þ h2yÞ

1=2: ð4:2Þ

Boundary conditions are

limy!1

h ¼ h1; limy!0

h ¼ 1; limy!0

p ¼ 0: ð4:3Þ

Integrating (4.1) and (4.2) once and using the boundary conditions (4.3) yields

h3 d2

dy2

hy

ð1þ h2yÞ

1=2¼ �3rbXðh� h1Þ þ ðh3 � h3

1Þ: ð4:4Þ

We rescale this equation to bring it into the form

h ¼ ðrbXÞ1=2�h; h1 ¼ ðrbXÞ1=2�h1; y ¼ ðrbXÞ1=6�y; ð4:5Þ
to get
�h3 d2

d�y2

�h�y

ð1þ ðrbXÞ2=3�h2�yÞ

1=2¼ �3ð�h� �h1Þ þ ð�h3 � �h3

1Þ: ð4:6Þ

For this equation, Wilson’s formula [8] gives the asymptotic approximation for the film thickness

�h1 ¼ 0:94581ðrbXÞ1=6;

i.e. from (4.5),

h1 ¼ 0:94581ðrbXÞ2=3: ð4:7Þ

Fig. 6 shows h1 as a function of rbX. Good agreement of the one-dimensional numerical results with thecorresponding higher order asymptotic formula is achieved for small values of rbX.

The meniscus profile h(y) in Fig. 7 is now computed for the values given in Section 4.1.1 Recall thatbX ¼ 1:089e� 3: At r = 9, we have rbX ¼ 0:009801, hence, from (4.7), h1 ¼ 0:0433.We now compare the meniscus profile computed with (4.6) for the drag-out problem with the numerical

solution for the steady state of our problems (2.41) and (2.45). For this we take results for the cross sectionalong constant radii. In Fig. 6 we performed the comparison for the cross section for the height profile atradius r = 9, for 0 6 h 6 180, i.e. from the point where the film is dragged out to the point where it reentersthe liquid bath. We see that there is excellent quantitative agreement in the vicinity of the meniscus region as itenters the thin film region.



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4.1.3. Comparison with the hyperbolic regime

Further out into the disk region the height profile will deviate from the height obtained for the drag-outproblem. There the variation of the height along the directions parallel to the disk is very small, which isclearly seen in our numerical simulations, so that surface tension will play a negligible role.

Starting from Eq. (2.41) we consider the steady state problem

Fig.

Fig. 7.(triang

PleaMod

o

orr

h3

3sin h

� �þ o

ohh3

3cos h� bXrh

� �¼ 0; ð4:8Þ

to describe the dynamics far away from the meniscus. This can be simplified to the hyperbolic equation

h2r sin hohorþ ðh2 cos h� bXrÞ oh

oh¼ 0: ð4:9Þ

Using the method of characteristics, this problem can be solved in form of an initial value problem for thesystem of the coupled ordinary differential equations

drds¼h2ðr0; 0Þr sin h; rð0Þ ¼ r0; ð4:10aÞ

dhds¼h2ðr0; 0Þ cos h� bXr; hð0Þ ¼ 0: ð4:10bÞ

0.10

0.1

0 0.2 0.4 0.6 0.8 1

rΩ ^

0

0.2

0.4

0.6

0.8

1

h8

Rescaled 1d numer. sol.Asymptotic solution

6. Comparison of numerical results for the one-dimensional problem (4.4) with the asymptotic formula for h1 versus rbX, (4.7).

5 10 15 20 25 30Arc length

0

0.1

0.2

0.3

0.4

h


Meniscus profiles computed with the 1D model (4.4) (curves with symbols), for X = 0.277 (circles), 1.0 (stars), 2.0 (plusses), 3.0les), and comparison with the profiles obtained for the cross section of the film profile for the rotating disk. ‘‘Arc length’’ denotes rh.


Fig. 8. Comparison of contour lines from our FEM computation (black curves) for X = 0.277 with the characteristics (white curves) forr0 = 1.962, 2.670, 3.560, 4.572, 5.759, 6.877 and corresponding heights of h0 = 0.0130, 0.0182, 0.0234, 0.0286, 0.0338, 0.0390, respectively.Note, the corresponding 3D plot in Fig. 4.


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Using as the initial condition the height found from (4.6) or simply by making use of formula (4.7) for a cho-sen r0 we can integrate (4.10a) and (4.10b) to obtain characteristics. This is shown in Fig. 8 for X = 0.277 as anexample. The results are similar for the other angular velocities. As can be seen, the comparison of the char-acteristics that start from the meniscus region shows good agreement with the contour lines found from theFEM computation. Note, that the contour lines that start at the boundary of the rotating disk strongly dependon the conditions there.

Interestingly, one can get a good idea on the film profile as a function of the angular velocity bX by simplysolving (4.10a) and (4.10b) for r as a function of h directly by taking

Fig. 9. Steady solution at X = 1 rpm for immersion depth of a = 0.2, 0.4, 0.6, 0.8.



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PleaMod

dr=dsdh=ds

¼ drdh¼ r sin h

cos h� bXr=h20

; ð4:11Þ

which can be solved to yield

rðhÞ ¼ h20bX cos h�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2 h� 2c0

bXh2

0

s0@ 1A; ð4:12Þ

where the integration constant is

c0 ¼ rð0Þ cos hð0Þ �bXr2ð0Þ

2h20

: ð4:13Þ

Fig. 10. Steady solution for immersion depth of a = 0.8 at angular velocity X = 0.277, 1.0, 2.0 rpm.


-10 -5 0 5 10x

0e+00

2e-04

4e-04h

a=0.2a=0.4a=0.6a=0.8

-10 -5 0 5 10x

0e+00

2e-04

4e-04

6e-04

h

Ω=0.277Ω=1.0Ω=2.0

Fig. 11. Horizontal cross sections through r = 0 for the film thickness as the immersion depth is increased (left) and as the angular velocityincreased (right).


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4.2. Film patterns for slightly immersed disks

When the disk is half-immersed (immersion height a = 0) the film thickness above the drag-out region willbe renewed with every rotation and can be described by the classic drag-out problem. Furthermore, the filmprofile is slightly deformed by the effects of gravity before being drawn into the liquid bath accompanied by atypical oscillation in the thickness of the film. The simple pattern emerging from this configuration changeswhen the immersion height is a is increased.

So in the final part of the paper we are interested in the new patterns that emerge when changing the twoparameters a and X.

In the following Fig. 9 we first show the effect of varying the immersion depth from a = 0.2 to a = 0.8, leav-ing all other parameters as in the case of the half-immersed disk with X = 1.0. We observe an emerging almostsymmetrical circular region where the film height has a minimum. This radius of the circular region increaseswith the immersion depth a. In fact, the radius quite closely corresponds to the distance from the minimum ofthe drag-in capillary wave to the vertical height of the rotation axis (a = 0, see Fig. 1). By adjusting a one cantherefore achieve thin and fairly constant film profiles.

When the angular velocity is increased, not only does the fluid volume on the disk increase but the circularregion is shifted towards the left, where the film is dragged out. This is demonstrated in Fig. 10 for the case ofthe immersion height a = 0.8. To see this effect more clearly we compare horizontal cross sections at r = 0 forthe film thickness, shown in Fig. 11.

5. Conclusions

In this work we have derived a dimension-reduced generalized lubrication model for the problem of thefully three-dimensional free boundary problem for the vertically rotating disk, dragging out a thin film froma liquid bath. The resulting two-dimensional nonlinear degenerate fourth-order boundary value problem wassolved numerically using a finite element scheme. For the steady state solutions we performed an asymptoticanalysis near the meniscus region and a careful comparison with cross sections of the numerical solutionsalong constant radii gave good agreement. Away from the liquid bath we could show good agreement ofour numerical solution with analytic solutions of the corresponding hyperbolic problem.

For slightly immersed disks we observed patterns for the film profile on the disk, which we studied as afunction of the immersion depth and the angular velocity. Interestingly, the dependence of surface area ofthe film profile and of the volume fluid dragged out is not trivial. Of course we only touched upon the richstructure of possible patterns and outlined some general tendencies. More detailed systematic study is subjectof current research. It is now possible to find optimum configurations that will be of importance for varioustechnological and industrial applications. For example, in PET-reactors the fluid mechanical problem is



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coupled to chemical reactions taking place mainly on the film surface, so that here it is important to obtainthin films with high surface area.

Acknowledgements

The authors acknowledge support by the DFG research center MATHEON, Berlin and the DFG grant WA1626/1-1. AM also acknowledges support via the DFG grant MU 1626/3-1.

References

[1] A.G. Emslie, F.T. Bonner, L.G. Peck, Flow of a viscous liquid on a rotating disk, J. Appl. Phys. 29 (1958) 858–862.[2] N. Fraysse, G.M. Homsy, An experimental study of rivulet instabilities in centrifugal spin coating of viscous newtonian and non-

newtonian fluids, Phys. Fluids 6 (1994) 1491–1504.[3] B. Reisfeld, S.G. Bankoff, S.H. Davis, The dynamics and stability of thin liquid films during spin coating. I. Films with constant rates

of evaporation or absorption, J. Appl. Phys. 70 (1991) 5258–5266.[4] T.G. Myers, J.P.F. Charpin, The effect of the coriolis force on axisymmetric rotating thin film flows, Int. J. Non-Linear Mech. 36

(2001) 629–635.[5] T.G. Myers, M. Lombe, The importance of the Coriolis force on axisymmetric horizontal rotating thin film flows, Chem. Eng.

Process. 45 (2006) 90–98.[6] B. Ozar, B.M. Cetegen, A. Faghri, Experiments on the flow of a thin liquid film over a horizontal stationary and rotating disk surface,

Exp. Fluids 34 (2003) 556–565.[7] L. Landau, B. Levich, Dragging of a liquid by a moving plate, Acta Physicochim. URSS 17 (1942) 42–54.[8] S.D.R. Wilson, The drag-out problem in film coating theory, J. Eng. Math. 16 (1982) 209–221.[9] L.W. Schwartz, On the asymptotic analysis of surface-stress-driven thin-layer flow, J. Eng. Math. 39 (2001) 171–188.

[10] A. Munch, The thickness of a Marangoni-driven thin liquid film emerging from a meniscus, SIAM J. Appl. Math. 62 (6) (2002) 2045–2063.

[11] M.S. Christodoulou, J.T. Turner, S.D.R. Wilson, A model for the low to moderate speed performance of the rotating disk skimmer, J.Fluids Eng. 112 (1990) 476–480.

[12] Bioengineering Institute. Fem/bem notes. The University of Auckland, New Zealand, 2005.[13] R. Reinelt, Delaunay gittergenerator Technischer Bericht, Hermann Fottinger-Institut, TU Berlin, 1995.[14] H.R. Schwarz, Methode der finiten elemente, Teubner Studienbucher, Stuttgart, 1991.[15] T.A. Davis, I.S. Duff, An unsymmetric-pattern multifrontal method for sparse lu factorization, SIAM J. Matrix Anal. Appl. 19 (1997)

140–158.[16] R.A. Cook, R.H. Clark, An analysis of the stagnant band on falling liquid films, Ind. Eng. Chem. Fundam. 12 (1973) 106–114.[17] H.S. Keshgi, S.F. Kistler, L.E. Scriven, Rising and falling film flows: viewed from a first order approximation, Chem. Eng. Sci. 47

(1992) 683–694.[18] K.J. Ruschak, Flow of a falling film into a pool, A.I. Ch. E.J. 24 (1978) 705–709.[19] S.D.R. Wilson, A.F. Jones, The entry of a falling film into a pool and the air-entrainment problem, J. Fluid. Mech. 128 (1982) 219–

230.


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